We recall that it was shown in [1]}, by applying the results of [2]}, [3]}, that
\(\mathcal {T}_{p,t_0}(z)=d(p,z)+t_0.\) 
If the initial time \(t_0\)  is zero, (or given) the knowledge of the arrival time function \(\mathcal {T}_{p,t_0}\)  on the open set \(\Gamma \subset \partial M\)  is equivalent to the corresponding travel time function \(r_p(\cdot )\)  as in (REF ).
